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Question Number 60756 by readone97 last updated on 25/May/19

$$expressinpartialfraction5/(x-2){(x+3)}^2$$

Commented by Prithwish sen last updated on 25/May/19

$$\frac{1}{(\mathrm{x}-2){(\mathrm{x}+3)}^2}=\frac{\mathrm{A}}{(\mathrm{x}-2)}+\frac{\mathrm{B}}{(\mathrm{x}+3)}+\frac{\mathrm{C}}{{(\mathrm{x}+3)}^2}$$$$\mathrm{and}\mathrm{then}\mathrm{proceed}.$$

Answered by ajfour last updated on 25/May/19

$$\frac{5}{(x-2){(x+3)}^2}=\frac{a}{x-2}+\frac{b}{x+3}+\frac{c}{{(x+3)}^2}$$$$a=\frac{5}{{(x+3)}^2}{\mid }_{x=2}=\frac{1}{5}$$$$c=\frac{5}{(x-2)}{\mid }_{x=-3}=-1$$$$nowinfirstlineletx=0$$$$\Rightarrow -\frac{5}{18}=-\frac{5}{2}+\frac{b}{3}-\frac{1}{9}$$$$\Rightarrow b=3\left(\frac{45+2-5}{18}\right)=7$$$$hence$$$$\frac{5}{(x-2){(x+3)}^2}=\frac{1}{5(x-2)}+\frac{7}{x+3}-\frac{1}{{(x+3)}^2}.$$

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